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158       XIII    INTEGRATION

z e K and u(z) < ^ u(z). (By the Hahn-Banach theorem, the points (z, ) belonging
to S are those with the following property: if h is any continuous affine linear function
on E x R such that /z(y, u(y)) ^ 0 for all y e K, then /z(z, ) ;> 0. (Use Problem 5.)

(b)    Show that if u is strictly convex in K, then u() < u(z) for all nonextremal points
z in K.

(c)    Show that, for each point z e K, there exists a positive measure ^ on K having z

as barycenter and such that u(z) == I u(y) <s^(y). (Apply Problem 2(b) to the closed

convex hull S of the compact set G in E x R.) Deduce from Problem 4 that u(z) = v(z)
almost everywhere in K, with respect to ft.

8.    Let E be a separable real Frechet space and K a compact convex subset of E.

(a)    Show that the set M of extremal points of K is the intersection of a sequence of
open sets of K, and therefore universally measurable. (In the space K x K x [0,1],
consider the set of triples (z', z", A) such that z' ^ z" and 0 < A < 1, and its image
K  M under the mapping (z', z"', A) i>Az' + (1  A)z".)

(b)    Prove that every point of K is the barycenter of a positive measure ft on K, with
total mass 1, such that ft(K  M) = 0 (" Choquef s theorem ") (Use Problems 6 and 7.)

9.    Extend the results of Problems 2 to 8 to the case of a convex and vaguely compact
subset K of M+(X), where X is a compact space (cf. Section 12.15, Problem 13).

11. THE SPACES L1 AND L2

Throughout the remainder of this chapter we shall make the following
convention: the function (x, y)\-+xy will be extended to R x R by putting
xy = 0 whenever one of the factors is 0 and the other is  oo (of course, this
extension to R of the multiplication on R is not continuous). With this
convention, we have x(yz) = (xy)z for all x, y, z<= R, and the relations
x ^ y and z ^ 0 imply xz <j yz.

If /is any mapping of X into K, we put

/**                                        / /**              \ 1/2

(13.11.1)         N1(/) =     \f\dft,       N2(/)

(1 3.11 .2)   For all mappings/ ^ 0 and g^O ofX into R, we have
(13.11 .2.1)          Np(/+ g) ^ Np(/) + Np(#)       (p = 1, 2).

This has already been proved for p = 1 (13.5.6). For p = 2 we shall first
prove (with the above convention for products in M) the generalized Cauchy-
Schwarz inequality

(13.11.2.2)                         Nx(/<7)  N2(/)N2(#).

This inequality is true if one of the factors on the right-hand side is zero.
If for example N2(#) =0, then g2 is negligible and therefore so is g (13.6.3)on K' with total mass 1